For the proof of Theorem REF  we employ a result of
Davenport and Schmidt [1]} related to Liouville's inequality [2]} on the distance
of two algebraic numbers. It states
that for any integer \(n\ge 1\)  and any real \(x\) , there is a constant \(c=c(n,\xi )>0\)  so that for any two integer polynomials
\(P,Q\)  of degree at most \(n\)  and without common factor,
if we let \(H=\max \lbrace  H_{P},H_{Q}\rbrace \)  then
\( \max \lbrace  |P(x)|, |Q(x)| \rbrace  \ge c H^{-(2n-1)}.\) 
